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Speculating on the Future of Mathematics
Appl. Math. Lett. Vol. 1, No. 1, pp. 79-82, 1988 Printed in the U.S.A. All rights reserved.
Copyright
0893-9659/88 (c) 1988 Pergamon
$3.00 + 0.00 Journals Ltd
Speculating on the Future of Mathematics Thomas L. Saaty University of Pittsburgh Modern mathematics continues to lag the real world by not offering new theories and concepts. Despite great excitement within the field about all sorts of progress in the subject, the and application of mathematics to new political and availability social structures is primitive and its relevance is interpreted only by a tour de force, not by philosophically tractable and justifiable It is one thing to start an idea and approaches. make it sound as if it has great promise and another to give it penetrating success. For example, Catastrophe Theory, born from pure mathematics without observation and experiment, has not lived up to the high expectations and mystique it generated. It is probably widely known that mathematics is dependent on the medium in which it is expressed: writing symbols and drawing pictures on some kind of surface. It is also dependent on and largely derives from the activity of the sense organ that dominates the mind: the eye - geometry. Even though its concepts may be abstract and lofty, the expression of these concepts is constrained to the medium. If it cannot be expressed that way, it is not mathematics. We use language to express and communicate ideas derived through reason. But reason leads to music and art and these use other media for communication. Animals are known to be capable of a modicum of reasoning, but certainly they do not use our language. Thus even though we cannot explain it adequately, subconscious mathematical creativity is not the same as the form of expression we communicate it with: language, symbols, and geometry. I suspect we all have a lot of ideas that escape because we lack an appropriate medium of communication; we are trapped to represent them in a way identical to that used by the Greeks.
The origin of speech has been localized in the brain to Broca's and Wernicke's areas both in the left lobe, the first controlling speech muscles and the second grammar. There is no such special part known in the left brain for quantitative thinking. It has been argued that the evolution of our species and those of some simian cousins has been guided by speech genetics. Thus while language, which is an intrinsic brain structure activity, is reducible to symbols, mathematics, which only in part uses language, probably relates to all neural firing and other precision workings of the brain and thus its expression could be broader and does not need a unique method of expression as does speech. Mathematics is a dimension and an outlet for imagination. It is our articulation of discrimination and feelings about intensities of properties, decompositions and aggregations of 79
80
T. L. SAATY
properties and relations among these as we perceive or experience them. Such feelings and ideas precede meaningful interpretations of the real world. concepts The availability of mathematical disposes us to interpret the kaleidoscopic changing of experience prepares in the most organized and latest way thus mathematics and organizes our thinking to interpret complex reality and to anticipate new states of that reality. Still, mathematics is not obligated to conform to the current popular view of experience Thus the which itself may be inconsistent and confused. development of mathematics is essential for gaining more detailed As we come to and deeper knowledge in the world of experience. our understand better how the brain works and develops, mathematics should both follow and guide the understanding of the brain, the psyche and how they can best affect the external environment. The most inhibiting aspect of medium-constrained mathematics is that we have not been able to deal with the details of structures in a faithful way because it is difficult or impossible to represent complex structures like living forms on a flat surface using symbols or smooth drawings. By assuming the absoluteness of the medium in defining our mathematics we are probably depriving ourselves of making revolutionary changes in mathematics is a social our field. Whereas communicating activity, creating it is an individual one. We owe it to ourselves to find alternative ways for both individual and social expression and not constrain too much the former by the needs of the latter. the notion of field, The world of feeling and perception, stimulus-response, social and political affection and interaction to symbols on may not be immediately or usefully transcribed They may need the manipulation of complex highly diverse paper. greater non-specific clusters of ideas or feelings with with all sorts of detail from macro to micro - not variability, not anything easily standard, not uniform, not homogeneous, tractable with our current mathematics. communicate What are all the means we might use to ideas and what is the effect of technology on these unambiguous means? How shall we go about expanding our media? Maybe we can use thought experiments, or gedankengeschichte as they are known, and scenarios that enable us to project so we can invent new ways to describe mathematical activity. A mathematician may say, "It is not up to me, I invent the tools. The scientist is free to use them as he pleases to understand his world." But is this in fact true? Look at what the 200 year old discovery of electricity has engendered in terms of concepts, problems, models and theories in The topology of surfaces owes a part of its origin mathematics. to differential equations relating to physics and astronomy. our for example, are suggested, assisted, maybe ideas in astronomy, even started by using a telescope to see into the universe. Ironically, the lenses of that telescope are ground and shaped Holography has precision. care and with mathematical revolutionized our thinking about how to record images on a
Speculating
on the future
of mathematics
81
photographic plate and how then to use them to recover the original image, transformed into three dimensions. Mathematical concepts may sometimes even lag behind concepts opened to us by technology. One way to overcome our static nature would be to use computers to bring long spans of time into our purview. More and the more we find that we are unable to retain and reconstruct Our assumptions circumstances which created our past thinking. change and hence our abstractions tend to assume our present We state of mind because that is what we are conscious of. cannot believe that we and others could be dramatically different from what we know now. Computers could serve as super cameras, super memories and super analyzers of us in different time and We need to collect the pieces in logical space settings. to synthesize outcomes for us which cut across time hierarchies In other words, we can use and space and people and conditions. the versatility of the computer to obtain a more objective image of ourselves. Hierarchic structures and feedback networks can play a useful role to develop this understanding. Geometry, algebra, calculus, topology, combinatorics and set theory, some of the fundamental mathematical systems we have developed so far, are largely amenable to or traceable in substance to vision or to motion - change of position or state. One can generate a sizeable list of problems which do not fit in any of these categories. For example, we have no adequate theory to deal with biological forms and the evolution of these forms. We need a theory of forms, what causes different forms, how they change, and how their underlying laws themselves change as they govern nature. We do not have an adequate theory to explain a possible field behavior of the brain. More simply, we have no good theory to deal with predicting the future. Geometry gives reasonable representations of visual images, but we have no intuitively understood symbolism for hearing. Acoustics is studied either abstractly or is represented through geometry, which makes more sense to the eye than to the ear. Only if sounds follow these curves will we believe that we understand, but this involves performance which is what we are trying to explain through abstract thinking, and is therefore unsatisfactory as a model. Another difference between seeing and hearing is that seeing usually has a much longer duration and memory record than does hearing, which is sequential in time for most of us. Similarly we have no satisfactory theories for smell, taste and touch so we can integrate these perceptions mathematically to give us a richer synthesis of our collective senses. This means that our understanding of ourselves is still relatively crude and suboptimal. I believe that there is a great challenge here to expand our modeling to deal with the reality of who we are first before we can deal adequately with the reality we experience in a satisfactory way.
82
T.L.
SAATY
are At present those who work in Artificial Intelligence striving to find new approaches by creating intelligent computers to solve problems for us. Our hope is that the results will and the match the talent, the resources spent, the euphoria, expectations built up in us. It may be optimistic to claim that mathematics is capable of We being a deep expression of a large part of human perception. can better realize its scope by varying its media, becoming aware of the limitations that the medium imposes on the growth of the subject, and overcoming these limitations with as much ingenuity raw , not well-structured as we show in solving very difficult, mathematical problems. It would be preferable if such a change were to take place with a conscious effort on our part. There are signs that it is happening anyway.
Copyright
0893-9659/88 (c) 1988 Pergamon
$3.00 + 0.00 Journals Ltd
Speculating on the Future of Mathematics Thomas L. Saaty University of Pittsburgh Modern mathematics continues to lag the real world by not offering new theories and concepts. Despite great excitement within the field about all sorts of progress in the subject, the and application of mathematics to new political and availability social structures is primitive and its relevance is interpreted only by a tour de force, not by philosophically tractable and justifiable It is one thing to start an idea and approaches. make it sound as if it has great promise and another to give it penetrating success. For example, Catastrophe Theory, born from pure mathematics without observation and experiment, has not lived up to the high expectations and mystique it generated. It is probably widely known that mathematics is dependent on the medium in which it is expressed: writing symbols and drawing pictures on some kind of surface. It is also dependent on and largely derives from the activity of the sense organ that dominates the mind: the eye - geometry. Even though its concepts may be abstract and lofty, the expression of these concepts is constrained to the medium. If it cannot be expressed that way, it is not mathematics. We use language to express and communicate ideas derived through reason. But reason leads to music and art and these use other media for communication. Animals are known to be capable of a modicum of reasoning, but certainly they do not use our language. Thus even though we cannot explain it adequately, subconscious mathematical creativity is not the same as the form of expression we communicate it with: language, symbols, and geometry. I suspect we all have a lot of ideas that escape because we lack an appropriate medium of communication; we are trapped to represent them in a way identical to that used by the Greeks.
The origin of speech has been localized in the brain to Broca's and Wernicke's areas both in the left lobe, the first controlling speech muscles and the second grammar. There is no such special part known in the left brain for quantitative thinking. It has been argued that the evolution of our species and those of some simian cousins has been guided by speech genetics. Thus while language, which is an intrinsic brain structure activity, is reducible to symbols, mathematics, which only in part uses language, probably relates to all neural firing and other precision workings of the brain and thus its expression could be broader and does not need a unique method of expression as does speech. Mathematics is a dimension and an outlet for imagination. It is our articulation of discrimination and feelings about intensities of properties, decompositions and aggregations of 79
80
T. L. SAATY
properties and relations among these as we perceive or experience them. Such feelings and ideas precede meaningful interpretations of the real world. concepts The availability of mathematical disposes us to interpret the kaleidoscopic changing of experience prepares in the most organized and latest way thus mathematics and organizes our thinking to interpret complex reality and to anticipate new states of that reality. Still, mathematics is not obligated to conform to the current popular view of experience Thus the which itself may be inconsistent and confused. development of mathematics is essential for gaining more detailed As we come to and deeper knowledge in the world of experience. our understand better how the brain works and develops, mathematics should both follow and guide the understanding of the brain, the psyche and how they can best affect the external environment. The most inhibiting aspect of medium-constrained mathematics is that we have not been able to deal with the details of structures in a faithful way because it is difficult or impossible to represent complex structures like living forms on a flat surface using symbols or smooth drawings. By assuming the absoluteness of the medium in defining our mathematics we are probably depriving ourselves of making revolutionary changes in mathematics is a social our field. Whereas communicating activity, creating it is an individual one. We owe it to ourselves to find alternative ways for both individual and social expression and not constrain too much the former by the needs of the latter. the notion of field, The world of feeling and perception, stimulus-response, social and political affection and interaction to symbols on may not be immediately or usefully transcribed They may need the manipulation of complex highly diverse paper. greater non-specific clusters of ideas or feelings with with all sorts of detail from macro to micro - not variability, not anything easily standard, not uniform, not homogeneous, tractable with our current mathematics. communicate What are all the means we might use to ideas and what is the effect of technology on these unambiguous means? How shall we go about expanding our media? Maybe we can use thought experiments, or gedankengeschichte as they are known, and scenarios that enable us to project so we can invent new ways to describe mathematical activity. A mathematician may say, "It is not up to me, I invent the tools. The scientist is free to use them as he pleases to understand his world." But is this in fact true? Look at what the 200 year old discovery of electricity has engendered in terms of concepts, problems, models and theories in The topology of surfaces owes a part of its origin mathematics. to differential equations relating to physics and astronomy. our for example, are suggested, assisted, maybe ideas in astronomy, even started by using a telescope to see into the universe. Ironically, the lenses of that telescope are ground and shaped Holography has precision. care and with mathematical revolutionized our thinking about how to record images on a
Speculating
on the future
of mathematics
81
photographic plate and how then to use them to recover the original image, transformed into three dimensions. Mathematical concepts may sometimes even lag behind concepts opened to us by technology. One way to overcome our static nature would be to use computers to bring long spans of time into our purview. More and the more we find that we are unable to retain and reconstruct Our assumptions circumstances which created our past thinking. change and hence our abstractions tend to assume our present We state of mind because that is what we are conscious of. cannot believe that we and others could be dramatically different from what we know now. Computers could serve as super cameras, super memories and super analyzers of us in different time and We need to collect the pieces in logical space settings. to synthesize outcomes for us which cut across time hierarchies In other words, we can use and space and people and conditions. the versatility of the computer to obtain a more objective image of ourselves. Hierarchic structures and feedback networks can play a useful role to develop this understanding. Geometry, algebra, calculus, topology, combinatorics and set theory, some of the fundamental mathematical systems we have developed so far, are largely amenable to or traceable in substance to vision or to motion - change of position or state. One can generate a sizeable list of problems which do not fit in any of these categories. For example, we have no adequate theory to deal with biological forms and the evolution of these forms. We need a theory of forms, what causes different forms, how they change, and how their underlying laws themselves change as they govern nature. We do not have an adequate theory to explain a possible field behavior of the brain. More simply, we have no good theory to deal with predicting the future. Geometry gives reasonable representations of visual images, but we have no intuitively understood symbolism for hearing. Acoustics is studied either abstractly or is represented through geometry, which makes more sense to the eye than to the ear. Only if sounds follow these curves will we believe that we understand, but this involves performance which is what we are trying to explain through abstract thinking, and is therefore unsatisfactory as a model. Another difference between seeing and hearing is that seeing usually has a much longer duration and memory record than does hearing, which is sequential in time for most of us. Similarly we have no satisfactory theories for smell, taste and touch so we can integrate these perceptions mathematically to give us a richer synthesis of our collective senses. This means that our understanding of ourselves is still relatively crude and suboptimal. I believe that there is a great challenge here to expand our modeling to deal with the reality of who we are first before we can deal adequately with the reality we experience in a satisfactory way.
82
T.L.
SAATY
are At present those who work in Artificial Intelligence striving to find new approaches by creating intelligent computers to solve problems for us. Our hope is that the results will and the match the talent, the resources spent, the euphoria, expectations built up in us. It may be optimistic to claim that mathematics is capable of We being a deep expression of a large part of human perception. can better realize its scope by varying its media, becoming aware of the limitations that the medium imposes on the growth of the subject, and overcoming these limitations with as much ingenuity raw , not well-structured as we show in solving very difficult, mathematical problems. It would be preferable if such a change were to take place with a conscious effort on our part. There are signs that it is happening anyway.